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cs-401r:homework-0.1 [2014/09/03 06:00]
ringger created
cs-401r:homework-0.1 [2014/12/31 23:17] (current)
ringger
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 === Question 1: Useful theorems in probability theory === === Question 1: Useful theorems in probability theory ===
-[50 points; 10 per subproblem]+[50 points; 10 per sub-problem]
  
 (Adapted from: Manning & Schuetze, p. 59, exercise 2.1) (Adapted from: Manning & Schuetze, p. 59, exercise 2.1)
  
 Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following five statements. ​ Develop your proof first in terms of sets and then translate into probabilities;​ use set theoretic operations on sets and arithmetic operators on probabilities. ​ Be sure to apply [[Proofs|good proof technique]]:​ justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification,​ you must first satisfy the conditions / pre-requisites of the axiom. ​   Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following five statements. ​ Develop your proof first in terms of sets and then translate into probabilities;​ use set theoretic operations on sets and arithmetic operators on probabilities. ​ Be sure to apply [[Proofs|good proof technique]]:​ justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification,​ you must first satisfy the conditions / pre-requisites of the axiom. ​  
-<​math>​P(B - A) = P(B) - P(A \cap B)</​math>​ +$P(B - A) = P(B) - P(A \cap B)$ 
-<​math>​P(A \cup B) = P(A) + P(B) - P(A \cap B)</​math> ​(the addition rule)+$P(A \cup B) = P(A) + P(B) - P(A \cap B)(the addition rule)
 #* Hint: refer to part result #1 as a part of your proof of part #2 #* Hint: refer to part result #1 as a part of your proof of part #2
-<​math>​P(\varnothing) = 0</​math>​ +$P(\varnothing) = 0$ 
-<​math>​P(\neg A) = 1 - P(A)</​math>​+$P(\neg A) = 1 - P(A)$
 #* Hint: refer to part result #1 as a part of your proof of part #4 #* Hint: refer to part result #1 as a part of your proof of part #4
-<​math>​A \subseteq B \Rightarrow P(A) \leq P(B)</​math>​ +$A \subseteq B \Rightarrow P(A) \leq P(B)$ 
-#* Hint: let <​math>​C = B - A</​math> ​(set difference).+#* Hint: let $C = B - A(set difference).
  
 === Question 2: Joint Probability === === Question 2: Joint Probability ===
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 * Let <​tt>​is-abbreviation</​tt>​ denote the event "this period indicates an abbreviation",​ and let <​tt>​three-letter-word</​tt>​ denote "a period occurs after a three letter word" * Let <​tt>​is-abbreviation</​tt>​ denote the event "this period indicates an abbreviation",​ and let <​tt>​three-letter-word</​tt>​ denote "a period occurs after a three letter word"
 * Assume the following probabilities:​ * Assume the following probabilities:​
-** <​math>​P(</​math>​<​tt>​is-abbreviation</​tt>​<​math> ​</​math>​<​tt>​three-letter-word</​tt>​<​math>​) = 0.8</​math>​ +** $P($<​tt>​is-abbreviation</​tt>​$<​tt>​three-letter-word</​tt>​$) = 0.8$ 
-** <​math>​P(</​math>​<​tt>​three-letter-word</​tt>​<​math>​) = 0.0003</​math>​+** $P($<​tt>​three-letter-word</​tt>​$) = 0.0003$
  
 === Question 3: Independence === === Question 3: Independence ===
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 (Adapted from: Manning & Schuetze, p. 59, exercise 2.4) (Adapted from: Manning & Schuetze, p. 59, exercise 2.4)
  
-Are <​math>​X</​math> ​and <​math>​Y</​math> ​as defined in the following table independently distributed?​+Are $Xand $Yas defined in the following table independently distributed?​
  
 <table border=1 cellspacing=0>​ <table border=1 cellspacing=0>​
 <tr> <tr>
-  <td><​math>​x</​math>​</td>+  <td>$x$</td>
   <​td>​0</​td>​   <​td>​0</​td>​
   <​td>​0</​td>​   <​td>​0</​td>​
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 </tr> </tr>
 <tr> <tr>
-  <td><​math>​y</​math>​</td>+  <td>$y$</td>
   <​td>​0</​td>​   <​td>​0</​td>​
   <​td>​1</​td>​   <​td>​1</​td>​
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 </tr> </tr>
 <tr> <tr>
-  <td><​math>​P(X=x, Y=y)</​math>​</td>+  <td>$P(X=x, Y=y)$</td>
   <​td>​0.32</​td>​   <​td>​0.32</​td>​
   <​td>​0.08</​td>​   <​td>​0.08</​td>​
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 [10 points] [10 points]
  
-# How many possible ways can you completely factor the joint distribution ​<​math>​P(X_1, X_2, X_3, X_4, X_5, X_6)</​math>​+# How many possible ways can you completely factor the joint distribution ​$P(X_1, X_2, X_3, X_4, X_5, X_6)$
-# For some arbitrary joint probability distribution on six random variables ​<​math>​P(X_1, X_2, X_3, X_4, X_5, X_6)</​math>​, apply the chain rule to completely factor this distribution in one way.+# For some arbitrary joint probability distribution on six random variables ​$P(X_1, X_2, X_3, X_4, X_5, X_6)$, apply the chain rule to completely factor this distribution in one way.
  
 === Question 5: Conditional Probability === === Question 5: Conditional Probability ===
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 [10 points] [10 points]
  
-Prove the chain rule, namely that <​math>​P(\cap^n_{i=1} A_i) = P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdot ...\cdot P(A_n | \cap^{n-1}_{i=1} A_i)</​math>​.  Justify each step.  Use the same standard of proof as in problem #1 above.+Prove the chain rule, namely that $P(\cap^n_{i=1} A_i) = P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdot ...\cdot P(A_n | \cap^{n-1}_{i=1} A_i)$.  Justify each step.  Use the same standard of proof as in problem #1 above.
 * Hint: you could use induction (but that isn't the only way). * Hint: you could use induction (but that isn't the only way).
  
cs-401r/homework-0.1.txt · Last modified: 2014/12/31 23:17 by ringger
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